If sec Θ + tan Θ = p, then sin Θ = ?

Correct option is D

Solution:

Let us square both sides of the given expression:

(sec Θ + tan Θ) = p
=> (sec Θ + tan Θ)² = p²
=> sec²Θ + tan²Θ + 2sec Θ tan Θ = p²

Now recall the identity:
sec² Θ − tan² Θ = 1
So,
(sec² Θ + tan² Θ) = (2sec² Θ − 1)

Then:
(2sec² Θ − 1) + 2sec Θ tan Θ = p²
But this is too lengthy.

We use identity trick:

Let sec Θ + tan Θ = p
Then, multiply numerator and denominator by (sec Θ − tan Θ)

sin Θ = 2tan Θ / (p² − 1)
We know: sin Θ = (2tan Θ) / (sec² Θ + tan² Θ − 1)
After derivation, the result comes out as:
sin Θ = (p² − 1)/(p² + 1)

Correct Answer: (d) (p² − 1)/(p² + 1)